It seems that a huge, rectangular shield like those stereotypical Roman shields seen in movies would be best for protection. If you lose less men, surely the extra cost of a bigger shield far outweighs dragging men in, training them, clothing them, outfitting them in armor (which can be taken off the dead bodies by the other side), etc.
Is there a reason for the smaller circles that seem so prevalent? I've seen them from England to China in my research but this doesn't make sense.
Shields aren't free protection; they come with multiple costs. In particular, while a large shield covers more, we have the following problems:
For a given thickness, it is heavier. Weight affects the speed of the shield in combat, stamina in combat, and mobility of the soldier in combat. A shield is most effective when it is mobile. While a small buckler 12" in diameter covers much less area than a 24" diameter medium-sized round shield - only 1/4 the area - it provides more than 1/4 the protection, especially against hand-to-hand attacks, since it can be moved to meet the attacks. A shield can provide active protection - it isn't just a thing to hide behind.
If one opts to make the shield larger, while keeping the weight the same, the shield becomes weaker, and attacks that are stopped by a smaller but thicker shield can penetrate the larger thinner shield.
Larger shields are larger and bulkier. A shield not only protects you against your opponent's attacks, but also gets in the way of your own attacks and blocks your own vision. A larger shield is a worse impediment to your own attacks, vision, and movement than a smaller shield.
A large shield is easier for the opponent to hook, press, or pull. It is a larger object with a larger rim, so it's easier for the opponent to engage it, and the greater distance from the grip to the rim offers the opponent more leverage to manipulate it. Corners can make this worse - a round shield is less susceptible to this kind of thing.
That we see so many large shields, whether 36" round shields, or large oval, rectangular, and kite-shaped shields, shows that large shields were appreciated. They are especially when they are intended for use by infantry who will fight in close formation. When are they less common?
Cavalry. A large shield is difficult to move around a horse's neck - cavalry shields are often smaller than infantry shields (about 24" diameter round shields are common).
Mobile infantry. Skirmishers and other infantry who don't fight in close formation often use smaller shields - their smaller shields are lighter and allow easier movement. Their smaller shields provide less protection against enemy archers/slingers, but they can use their mobility to stay out of range.
Civilians. A large shield is a heavy and bulky item to carry around all day. Very small shields (bucklers) or no shield at all are common options for civilians. Apart from the light weight and convenient small size, bucklers provide good protection against a single opponent in hand-to-hand combat. Missile weapons are less likely to be encountered by civilians than by soldiers on a battlefield - they have less need to stop arrows and the like.
Note that when two-handed polearms (or guns, or crossbows, or bows) become the most common infantry weapons on the battlefield, the majority of shields are cavalry shields or civilian bucklers. Guns also have major effects on shields - when guns are common enough, shields either disappear or are made thicker to resist bullets which necessitates becoming smaller in order to keep the weight manageable.
A shift to two-handed weapons as the main infantry weapons means that something other than a shield provide the necessary protection against weapons such as guns, bows, and crossbows. Sometimes, this is provided by armour, and sometimes this is provided by tactics (e.g., using your own archers/musketeers to keep enemy archers/musketeers out of effective range of your pikemen).
Some even larger shields are used - much larger than stereotypical Roman shields. For example, pavises and mantlets. Such very large shields are too large to be used in hand-to-hand combat, and function as portable fortifications.
Edit:
To see a Medieval European buckler in use for active parrying/blocking, trapping, and striking, see Talhoffer's fightbook: http://wiktenauer.com/wiki/Hans_Talhoffer/W%C3%BCrttemberg (sword and buckler section).
For a comparison between small and large shields (buckler and target), we have George Silver's comments:
First,
The sword and buckler has advantage against the sword and target,
but the target has some advantages:
Yet understand, that in battles, and where variety of weapons are, among multitudes of men and horses, the sword and target, the two handed sword, battle axe, the black bill, and halberd, are better weapons, and more dangerous in their offense and forces, than is the sword and buckler, short staff, long staff, or forest bill. The sword and target leads upon shot, and in troops defends thrusts and blows given by battle axe, halberds, black bill, or two handed swords, far better than can the sword and buckler.
Why the buckler is better, one-on-one:
The sword & target together has but two fights, that is the variable fight, & the close fight, for the close fight, the number of his feet are too many to take against any man of skill having the sword & buckler, & for the variable fight although not so many in number, yet too many to win the place with his foot and strike home. The sword & buckler man out of his variable, open & guardant fight can come bravely off & on, false & double, strike & thrust home, & make a true cross upon every occasion at his pleasure. If the sword & target man will fly to his guardant fight, the breadth of the target will not suffer it, if to his open fight, then has the sword & buckler man in effect the sword and buckler to the single, for in that fight by reason of the breadth, the target can do little good or none at all.
Essentially, the large shield gets in the user's way. For Silver's work: http://wiktenauer.com/wiki/George_Silver
Finally, an interesting type of shield that combines some of the advantages of buckler and large shield: the long narrow parrying shield. It still has the disadvantage of only having a small area to protect against arrows and the like, but provides a long length for parrying/blocking hand-to-hand weapons, while remaining light and mobile and minimally blocking vision. Some examples:
Sudanese: https://www.pinterest.com/pin/445363850630634547/
Indian: https://i.pinimg.com/originals/d3/2f/ab/d32fab91781150b5366ca51bd0818a5a.jpg
